Travel to the Great Khan Encampment and inquire about why Melissa is there to begin the quest. If you have bad reputation with the Great Khans she will refuse your offer of help. If vilified, wear Great Khan armor and helmet and the quest will be available to you. It is possible that Melissa may also be found in Quarry Junction.
Travel to Sloan and enter the mining office. Walk up the stairs and examine the empty suitcase to the left of the desk.
Go outside and speak with Chomps Lewis and inquire about the empty suitcase to find out about Tyrone. If Lewis is killed prior to this quest, the marker will lead to his body. There you can find a letter to Tyrone. After reading it, this step is completed.
Travel to Primm and find Tyrone at the NCR camp in the tent next to Lieutenant Hayes' tent.
When you question him about the chems, he explains that his superiors are getting suspicious and that the business is riskier. He will ask for an additional 300 caps which can be paid to obtain the chems.
The chemicals can be obtained outright with no penalties with any of the following:
You can just shoot Tyrone and take the chems, but that will result in NCR infamy and negative karma.
Once the chems have been obtained return to the Great Khan Encampment and give the chems to Melissa.
(Optional) You can inform Lieutenant Hayes in his tent about Tyrone's smuggling. If done, Hayes will take the chems and inquire if anyone else was involved. Chomps Lewis can be ratted out or Hayes can be told nobody else was involved. After retaking the NCRCF from the Powder Gangers, Lt. Hayes will have been relocated to Camp Forlorn Hope, where he will stand by the campfire opposite the supply shack.
Giving the chems to Melissa earns fame with the Great Khans, 100XP, 150 caps, and some chems.
Helping Lieutenant Hayes earns fame with the NCR, increased Karma, and 100XP.
Returning to Melissa, after informing Hayes about the smuggling, results in no reward. If claimed, through a lie, that the player doesn't know anything, or if the player simply claims that Tyrone was sloppy, Melissa will complain then claim to return to Red Rock Canyon without fighting the player. If the player admits that they turned in Tyrone, Melissa and the other two Great Khans will turn hostile.
Search the suitcase at Sloan for the chemical supply shipment.
Check with Chomps Lewis about the empty suitcase in his office.
Acquire the chem shipment from Tyrone in Primm.
(Optional) Expose the Great Khan smuggling operation to Lieutenant Hayes.
Return to Melissa with the chem shipment.
Behind the scenes
Don't Make a Beggar of Me is the name of a song written by Al Sherman and arranged by Axel Stordahl. It was recorded by Frank Sinatra in 1953 but was unreleased until late 2000 when Sinatra's 1962 album Point of No Return was re-released on CD. Don't Make a Beggar of Me was one of the bonus tracks on this CD.